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Reflection of caustics and focused sonic booms
Wave Motion 42 (2005) 211–225
Reflection of caustics and focused sonic booms Pablo Luis Rend´on ∗ , Franc¸ois Coulouvrat Laboratoire de Mod´elisation en M´ecanique, UMR CNRS 7607, Universit´e Pierre et Marie Curie, case 162, 4 Place Jussieu, 75252 Paris Cedex 05, France Received 23 November 2004; received in revised form 2 February 2005; accepted 8 February 2005 Available online 5 March 2005
Abstract Sonic boom, a community acceptance problem for the future development of civil high speed aircraft, is most intense (the so-called “superboom”) when focussed during the aircraft transsonic acceleration from Mach 1 to cruise speed. The ground reflection of superboom is studied. It is shown that the incident and the reflected fields are solutions of two uncoupled nonlinear Tricomi equations. The coupling between the two fields is expressed straightforwardly only through a linear ground impedance condition. For waves reflected off the ground or off rough sea surface, the effects of ground impedance may be noticeable, especially in the case of a soft ground. Amplitude reduction is also enhanced in the weakly nonlinear case, where the peaks are sharper. That behaviour is similar for reflection off a sea rough surface. However, in all investigated cases, the amplitude reduction of the reflected field remains limited. Hence, from a perception point of view, no significant mitigation of superboom is likely to be expected from passive absorption. © 2005 Elsevier B.V. All rights reserved. Keywords: Nonlinear acoustics; Sonic boom; Focusing; Caustics; Ground effects; Impedance
Introduction Sonic boom is a community acceptance problem that remains to be overcome for a supersonic overland flight, a key point for the future development of any profitable and green civil high speed aircraft. The most intense sonic boom, the so-called superboom due to sound focusing occuring during the aircraft transsonic acceleration from Mach 1 to cruise speed, cannot be avoided by realistic maneuvers. For present aircraft configurations, it leads to an amplification of ground pressures up to two to three times the unfocused boom shock strength. To comply with a future international regulation on sonic boom, an accurate prediction of the level of focused booms is required. A new algorithm has been recently designed [1,2] that is numerically solving the nonlinear Tricomi [3,4] ∗ Corresponding author. Present address: Centro de Ciencia Aplicada y Desarrollo Tecnol´ ogico, Universidad Nacional Aut´onoma de M´exico, Ciudad Universitaria, M´exico, D.F. 04510, Mexico. Tel.: +5255 56228627x1218; fax: +5255 56228675. E-mail addresses: [email protected] (P.L. Rend´on), [email protected] (F. Coulouvrat).
0165-2125/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2005.02.002
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equation modelling locally the nonlinear sound field around a fold caustic (the simplest caustic in the hierarchy of catastrophe theory [5–8]). Laboratory-scale experimental simulations [9] provide quantitative validation of the model, in agreement with outdoor recordings during test flights [10,11]. However, the present models consider only caustics in the free field, while sonic boom annoyance is mostly at the ground or sea level. Until now, most calculations of the ground total field simply consider it to be twice the incident field on the ground or sea surface. While clearly convenient, the accuracy of this method would only seem guaranteed in the case of purely linear propagation and a rigid reflection surface. However, sonic boom shock waves are nonlinear, and numerical simulations show that absorbing properties of the ground may significantly influence sonic boom, mostly by increasing the rise time [12]. The question therefore remains open to determine the influence of ground properties on superboom, which amounts to modelling the problem of nonlinear reflection of caustics over a surface of finite impedance. In the present paper we propose an analytical model for the aforementioned reflection, both in the linear and the nonlinear cases, arrived at through a multiple scales method. Firstly, the geometry of the problem is presented before recalling the linear solution for the free field caustic described in the frequency domain in terms of Airy functions. In a second part, the caustic reflection is modelled in the nonlinear regime by the method of multiple scales, the mirror reflection in the linear, rigid case being used as an “Ariane thread” to sort out the intricate geometry of the problem. Despite this complexity, the result is outstandingly simple: even in the nonlinear case, both incident and reflected field satisfy two independent nonlinear Tricomi equations, the reflected field being coupled to the incident one only through the boundary condition at the ground or sea surface. This one is explicited in the third section for a surface with impedance, thus showing that the ground pressure field can be deduced from the nonlinear incident one simply by applying a linear reflection coefficient. Finally, examples of numerical simulations are performed to quantify the influence of ground porous absorption [13] or sea surface roughness [14]. 1. The geometry of the caustic Consider the two-dimensional problem of a fold caustic in a homogeneous, inviscid fluid with ambient wave speed c0 and density ρ0 . We only consider here the case of a concave caustic. This caustic arrives at a reflecting surface, either the ground or the sea surface, at an angle Ω; the sea or ground are characterised by complex impedance functions particular to each case, which will be discussed later in Sections 4 and 5. Locally, the caustic is be considered to be, essentially, a parabola, with the origin set at the point where the caustic runs into the ground or sea surface. We take an X-axis normal to the reflecting surface and pointing towards the fluid, and a Z-axis which runs along this same surface with the same orientation as sound propagation as is illustrated in Fig. 1. The origin is chosen at the contact point of the caustic with the ground.
Fig. 1. Geometry of the caustic reflection.
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For this particular geometry, the form of the incident pressure field in the vicinity of a caustic is given by Pierce [15] in the linear and high frequency approximation: pi (x, δ, t) ≈ K exp{i[ω0 t − kz − 4ε3 k2 xz + O(ε6 )]} Ai[2εkx + O(ε4 )],
(1)
with ω0 is the angular frequency, k = ω0 /c0 the wave number and Ai is the Airy function, which is the generic function describing the diffraction catastrophe for fold caustic according to catastrophe theory [5–8]. The x-axis points towards the direction normal to the caustic and the shadow zone (therefore towards ground as the caustic is concave) and the z-axis is tangent to the caustic, oriented along acoustical rays. Introducing the radius of curvature R of the caustic at the origin, the parameter ε = 1/(4kR)1/3 is a small diffraction parameter associated to the order of magnitude of diffraction effects near the caustic. In the coordinate system adapted to ground reflection x = −Z sin Ω − X cos Ω, and z = −Z cos Ω − X sin Ω, Eq. (1) now runs: ˜ Z, ˜ t) ≈ K exp{i[ω0 t − Z ˜ cos Ω + X ˜ sin Ω + 4ε3 cos 2Ω X ˜Z ˜ + 2ε3 sin 2Ω(Z ˜2 −X ˜ 2) pi (X, ˜ sin Ω + X ˜ cos Ω) + O(ε4 )], +O(ε6 )]} Ai[−2ε(Z
(2)
˜ = kX and Z ˜ = kZ have been scaled according to wavelength. where the variables X When considering reflection off a rigid ground, we may use the method of images to obtain the reflected pressure field. This is typically done by adding the contribution of a caustic placed in such a way as to be the exact mirror image with respect to the ground of the initially given incident caustic. To each ray tangent to the original caustic there will then correspond a single ray tangent to the mirror-image caustic that will satisfy the reflection condition at the ground, as is seen in Fig. 1. ˜ Z, ˜ t), The contribution to the pressure field by the original caustic, representing the incident rays, is given by pi (X, ˜ ˜ ˜ ˜ while the contribution of the image caustic is given by pr (X, Z, t) = pi (−X, Z, t). The total pressure field in the vicinity of the origin is then: ˜ Z, ˜ t) = pi (X, ˜ Z, ˜ t) + pi (−X, ˜ Z, ˜ t). p(X,
(3)
2. The method of multiple scales and the field equation The obtention of a governing equation for the pressure field in the vicinity of a caustic in free space relies on the correct choice of scalings using the nondimensional parameter ε to arrive at a compact form of this equation. Considering the reflection of this wave field off a certain surface makes things much less straightforward than in that previous case. Not only must we deal with the fields associated with incident rays and those reflected off the ground, but, more importantly, the arbitrary introduction of this boundary affects the geometry of the problem in ˜ and Z, ˜ as it was such a way that it is no longer possible to find natural scalings for each of the newly defined axes, X for the caustic in free space. In fact, attempts to find such scalings indicate that a multiple scales approach [16] is required to obtain governing equations unaffected by the influence of secular terms. In this section we will then use the method of multiple scales to obtain the governing equation for the pressure field in the vicinity of the caustic, starting from the nonlinear wave equation [17]: ∂2 Φ ∂ 1 B ∂Φ 2 2 2 , (4) − c0 ∆Φ = + (∇Φ) ∂t 2 ∂t c02 2A ∂t the gradient operator and is with Φ is the acoustical potential, B/A the parameter of nonlinearity of the fluid, ∇ the Laplacian. We assume that two scales are necessary to describe variations along the direction normal to the ground, X. The ˜ is associated solely to wavelength, while the second, X, ˆ will first of these, which we have already introduced, X, be scaled also according to the small parameter ε we have previously introduced. As we have seen in the previous
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section, the linear field associated with rays reflected off a rigid ground may be obtained from the incident field simply by replacing X by −X. When choosing the form of the phase, however, this poses a difficulty since a term proportional to X is present here, and should thus change sign depending on whether it is associated to the incident or the reflected field. We will then initially omit any terms proportional to X from the phase term, and reinstate them later when it will seem natural to do so. We start then by writing the phase as ˜ cos Ω + 2ε3 sin 2Ω(Z ˜2 −X ˜ 2 ), τˆ = ω0 t − Z
(5)
and the rest of the variables as ˆ = εkZ, Z
˜ = kX, X
ˆ = εkX, X
(6)
which is sufficient to express the total field in the linear, rigid case as ˜ Z, ˜ t) ≈ K exp[ˆτ + X ˜ sin Ω + 4ε cos 2Ω X ˆ Z] ˆ Ai[−2(Z ˆ sin Ω + X ˆ cos Ω)] p(X, ˜ sin Ω − 4ε cos 2Ω X ˆ Z] ˆ Ai[−2(Z ˆ sin Ω − X ˆ cos Ω)]. +K exp[ˆτ − X
(7)
Notice that it is not necessary to introduce two scales for the Z coordinate, as the fast variations in this direction are included in the part of the phase variable, Eq. (5) which is invariant under mirror reflection. In terms of these variables, we obtain the following relations: ∂ ∂ = ω0 , ∂t ∂ˆτ ∂ ∂ ∂ ˆ ∂ , = k −cos Ω + ε + 4ε2 k sin 2ΩZ ˆ ∂Z ∂ˆτ ∂ˆτ ∂Z ∂ ∂ ∂ ˆ ∂ , =k +ε − 4ε2 sin 2ΩX ˜ ˆ ∂X ∂ˆτ ∂X ∂X
(8) (9) (10)
ˆ in order to include the maximum number of terms of order up to where we have chosen to write ∂/∂X in terms of X ε2 in the Laplacian, which is then given by 2 2 2 ∂ Φ ∂ Φ ∂2 Φ −2 2 ∂ Φ + cos Ω 2 + 2ε − cos Ω k Φ = ˜2 ˜ X ˆ ˆ ∂ˆτ ∂X ∂X∂ ∂ˆτ ∂Z 2 ∂2 Φ ∂2 Φ ∂2 Φ 2 ∂ Φ ˆ ˆ +ε + O(ε3 ). + − 8 sin 2Ω X (11) + Z cos Ω 2 ˜ ˆ2 ˆ2 ∂ˆτ ∂ˆτ ∂X ∂X ∂Z The nonlinear terms are then rewritten as 2 2 1 B ∂Φ 2 B ∂Φ ∂Φ 2 2 2 + O(ε). + (∇Φ) = k + + cos Ω 2 ˜ 2A ∂ˆτ ∂X c0 2A ∂t
(12)
ˆ 0 where U0 is the amplitude of the acoustical velocity, and assume that We scale the potential by Φ = U0 c0 Φ/ω nonlinear effects are sufficiently small so that βM = U0 /c0 = O(ε2 ) where M is the acoustical Mach number and β = 1 + B/2A. This assumption is well satisfied for sonic boom applications: for a supersonic transport one has typically unfocused boom at the ground level of duration 0.2–0.3 s (= 1/ω0 ) and amplitude up to 120 Pa, thus leading to a parameter βM of the order of 10−3 , while the parameter ε2 would be about 4 × 10−3 for the standard atmosphere (with R about 100 km) and hence µ equal to 0.06. However, for realistic situations, the radius of curvature R is highly sensitive to meteorological conditions and can fluctuate between a few tens to several thousands of kilometers.
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We now propose a power series expansion of the form: ˆ 1 (ˆτ , X, ˆ =Φ ˆ 0 (ˆτ , X, ˆ 2 (ˆτ , X, ˜ X, ˆ Z) ˆ + εΦ ˜ X, ˆ Z) ˆ + ε2 Φ ˜ X, ˆ Z) ˆ + ···, Φ
(13)
and, in the context of a multiple-scales expansion, we seek to eliminate secular terms as they appear when matching terms of the same order in ε. We start by equating all terms of order ε0 to zero. The first equation we thus obtain is simply the wave equation: sin2 Ω
ˆ0 ˆ0 ∂2 Φ ∂2 Φ − = 0, ˜2 ∂ˆτ 2 ∂X
(14)
ˆ 0i (ξ, X, ˆ 0r = Φ ˆ 0r (η, X, ˆ 0i = Φ ˆ Z), ˆ with ξ = τˆ + X ˜ sin Ω, and Φ ˆ Z), ˆ with η = τˆ − X ˜ sin Ω; which has solutions Φ these solutions correspond, as indicated by the subindices, to the incident and reflected fields, respectively. Now, as the distinction between incident and reflected fields has become explicit, we reintroduce the terms which we omitted from the phase τ˜ , with the appropriate signs, as follows: ˜ sin Ω + 4εX ˆZ ˆ cos 2Ω, ξ = τˆ + X
˜ sin Ω − 4εX ˆZ ˆ cos 2Ω. η = τˆ − X
(15)
Notice that because the extra terms are of order ε only, Eq. (14) of order 1 and its solutions remain unchanged. The Kuznetsov equation at the order ε can be written: ˆ1 ˆ 0i ˆ 0r ˆ 0i ˆ 0r ∂2 Φ ∂ ∂Φ ∂Φ ∂Φ ∂ ∂Φ 2 = sin Ω sin Ω − cos Ω − + cos Ω , (16) 2 sin Ω ˆ ˆ ˆ ˆ ∂ξ∂η ∂ξ ∂η ∂X ∂Z ∂X ∂Z and integrating twice:
ˆ 0i ˆ 0i ∂Φ ∂Φ − cos Ω 2 sin ΩΦ1 = η sin Ω ˆ ˆ ∂X ∂Z
2
ˆ 0r ˆ 0r ∂Φ ∂Φ − ξ sin Ω + cos Ω ˆ ˆ ∂X ∂Z
ˆ 1i (ξ)+Φ ˆ 1r (η)). + 2 sin2 Ω(Φ (17)
It is at this point that the flexibility afforded by the introduction of multiple scales becomes apparent: it is possible to prevent two secular terms from appearing above, by requiring that ˆ 0i ˆ 0i ∂Φ ∂Φ − tan Ω = 0, ˆ ˆ ∂Z ∂X
and
ˆ 0r ˆ 0r ∂Φ ∂Φ + tan Ω = 0. ˆ ˆ ∂Z ∂X
(18)
Thus, we see that the solutions now have the form: ˆ 0i (ξ, u) ˆ 0i = Φ Φ
ˆ 0r = Φ ˆ 0r (η, v), and Φ
(19)
with ˆ sin Ω + X ˆ cos Ω) = −εkx, u = 2(Z
and
ˆ sin Ω − X ˆ cos Ω). v = 2(Z
(20)
Notice that u and v represent the distances to the planes tangent to the incident caustic and to the mirror caustic, respectively, and that, in the case of a concave incident caustic, we choose to have positive u and v both point towards the concave sides of the respective caustics, in the region occupied by acoustic rays. We point out, however, that u and v do not constitute an orthogonal coordinate system, as is shown in Fig. 1. The coefficient two is introduced for convenience. In similar fashion, to the first order we have ˆ 1i (ξ, u, v) + Φ ˆ 1r (η, u, v). ˆ1 =Φ Φ
(21)
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Rewriting the Kuznetsov Eq. (4) in terms of new variables (ξ, η, u, v) and injecting the asymptotic expansions Eqs. (13), (19) and (21) yields: 2ˆ ˆ 1i ˆ 1r ˆ 0i ˆ 0r ˆ 0i ˆ 0r ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ 2 ∂ Φ2 4 sin Ω + 4 sin Ω cos Ω + − +u +v +4 − ∂ξ∂η ∂ξ∂v ∂η∂u ∂u2 ∂v2 ∂ξ 2 ∂η2 2 2 2 ˆ 0i ˆ 0r ˆ 0i ∂Φ ˆ 0r ˆ 0r ˆ 0i ∂2 Φ M ∂ ∂Φ ∂ ∂Φ B ∂ Φ ∂Φ . = 2 β (22) +β +2 + cos 2Ω + ε ∂ξ ∂ξ ∂η ∂η 2A ∂ξ 2 ∂η ∂ξ ∂η2 Integrating twice with respect to variables (ξ, η) provides the following expression: 2 ξ 2ˆ ˆ ˆ ˆ ∂ Φ0i ∂Φ0i βM ∂Φ0i ˆ 2 = 4η −sin Ω cos Ω ∂Φ1i + dξ − u + 2 4 sin2 ΩΦ 2 ∂v ∂u ∂ξ 4ε ∂ξ −∞ ˆ 1r ∂Φ +4ξ −sin Ω cos Ω + ∂u
η
−∞
ˆ 0r ˆ 0r ∂2 Φ ∂Φ βM dη − v + 2 2 ∂v ∂η 4ε
ˆ 0r ∂Φ ∂η
2 + NST.
(23)
where the missing terms NST are non-secular terms. Now requiring that secular terms proportional to ξ and η both vanish, and integrating now in terms of variables u and v yields: 2 ξ 2ˆ ˆ 0i ˆ 0i Φ ∂ ∂ Φ ∂ Φ βM 0i + NST, ˆ 1i = v sin Ω cos ΩΦ dξ − u + 2 2 ∂ξ 4ε ∂ξ −∞ ∂u ˆ 1r = u sin Ω cos ΩΦ
η
−∞
ˆ ∂2 Φ
0r
∂v2
ˆ 0r ∂Φ βM dη − v + 2 ∂η 4ε
ˆ 0r ∂Φ ∂η
2 + NST.
The cancellation of secular terms, in this case proportional to u or v requires: 2 2 ˆ 0i ˆ 0i ˆ 0i ˆ 0i ˆ 0r ˆ 0r ∂2 Φ ∂2 Φ βM ∂ ∂Φ ∂2 Φ ∂2 Φ βM ∂ ∂Φ −u + 2 = 0, −v + 2 = 0. ∂u2 ∂ξ 2 4ε ∂ξ ∂ξ ∂v2 ∂η2 4ε ∂η ∂η
(24)
(25)
Thus, we are able to obtain uncoupled equations for the incident and reflected fields, as we expected, since interaction with the reflected waves is not, as it were, felt by the incident waves, and vice versa, even in the nonlinear case. Physically, this comes from the fact that the incident and reflected fields propagate in two different directions, so that their nonlinear interaction is not resonant, as is well known: the last term in Eq. (22) which is the only one ˆ ˆ = ∂Φi∂ξ(ξ,u) , where the incident and reflected fields do interact, is not secular. Now substituting pˆ i (ξ, u) = ∂Φi∂ˆ(ξ,u) τ and pˆ r (η, v) =
ˆ r (η,v) ∂Φ ∂ˆτ
=
ˆ r (η,v) ∂Φ ∂η ,
we may write:
∂2 pˆ i ∂2 pˆ i ∂2 (pˆ i )2 − u + µ = 0, ∂u2 ∂ξ 2 ∂ξ 2
∂2 pˆ r ∂2 pˆ r ∂2 (pˆ r )2 − v + µ = 0, ∂v2 ∂η2 ∂η2
(26)
with µ = βM/4ε2 is a nonlinear parameter measuring the relative magnitude order of nonlinear effects (βM) compared to diffraction ones (ε2 ). We thus obtain uncoupled nonlinear Tricomi equations for the incident and reflected fields in the region near the point where the caustic meets the ground. Guiraud [3] has previously shown that the Tricomi equation describes the pressure field in the vicinity of a caustic in free space, so this result is, in fact, an expected one, but is still important since it indicates that it is possible to deal with the case of the incident and the reflected fields
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separately, as the governing equations are uncoupled. The relation between the solutions to these two equations will be expressed through the boundary conditions imposed on them, and in particular an impedance condition given on the ground, as detailed in the next section. Finally, it should be noted that the present study is not applicable to the case of a grazing caustic. Indeed, for very small angles Ω, the first term in the left-hand side of Eq. (14) may turn out to be of second order ε, thus upsetting the present hierarchy of equations. Consequently, the present study is limited to grazing angles Ω larger than ε1/2 . In practice, for the focused sonic (super) boom resulting from transonic acceleration, the caustic intersection with the ground has typically the shape of a crescent. The center of this crescent is associated to focused boom emitted right below the aircraft (ground track focusing) at the smallest Mach numbers, while its extremities are grazing above the ground, and continuously match the caustics with the shadow zone [18]. Such a complex region is beyond the scope of the present study and would need further asymptotic modelling. It is to be noted that the limiting value Ω ≈ ε1/2 delimiting the validity of the present theory is not so small: for the typical value ε2 = 4 × 10−3 in the standard atmosphere, this provides a grazing angle necessarily larger than 14◦ .
3. The boundary conditions Two sets of boundary conditions are required, one set for the incoming field and another for the reflected field, so that, in total, four conditions must be given, due to the fact that the Tricomi equations obtained above are of second order for each of the two variables involved. Let us consider first the boundary conditions for the incoming field, which have the same form as the analogous expression for a free-field fold caustic [1–4]. For a transient signal, we require that the field vanishes for very large times, thus obtaining the two boundary conditions for ξ: pˆ i (ξ → ±∞, u) = 0.
(27)
Recall that the Tricomi equation is a mixed hyperbolic/elliptic equation. For u − µpi > 0, it is of hyperbolic type, and for u − µpi < 0, it is of elliptic type; thus, two different behaviours are to be expected on either side of the caustic. As in the case of a caustic in free space [1–4], on the elliptic side the field vanishes exponentially as we go away from the caustic and into the shadow zone. Since u indicates the direction normal to the caustic at the origin, this condition is here expressed as follows: pˆ i (ξ, u → −∞) −→ 0. exp
(28)
We also require that the pressure field matches the geometrical acoustics approximation far away from the caustic on the hyperbolic side of the caustic: pˆ i (ξ, u → +∞) ≈ |u|−1/4 [F (ξ − 23 u3/2 ) + G(ξ + 23 u3/2 )],
(29)
where the minus sign corresponds to F, the ingoing field before it tangents the caustic, and the plus sign corresponds to G, the outgoing field after it has tangented the caustic. In Eq. (29), only the ingoing field is known contrarily to the outgoing one. This last one can be eliminated so as to rewrite Eq. (29) under the form of a “radiation” condition involving only the incoming field [1,2,4]. Now we consider the boundary conditions for the reflected field. The condition regarding the phase is repeated for the reflected field: pˆ r (η → ±∞, v) = 0,
(30)
and so is the condition of exponential decay, applied, as before, to the field for large v, since the influence of the ground is not yet felt: pˆ r (η, v → −∞) −→ 0. exp
(31)
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However, for v positive, the influence of the ground on the propagation of the reflected rays must be taken into account to formulate the remaining condition. Assuming that the ground surface S is not rigid or impenetrable, we use a specific acoustic impedance [15] denoted ZS to characterize the interaction of the incoming acoustic rays with the ground, and which is a function only of the angular frequency ω and the angle Ω of incidence of the incoming field, and is given in this case by TF[p](X, Z, ω) , (32) ZS (ω, θi ) = c0 ρ0 ζS (ω, θi ) = − TF[VX ](X, Z, ω) X=0 where VX (X, Z, t) is the fluid velocity in the ground direction. The ratio of the specific impedance ZS to the characteristic air impedance is denoted ζ S , a dimensionless impedance which we will use in what follows. The direct and inverse Fourier transforms are defined, for dimensional, physical variables, as follows: ∞ ∞ 1 TF[p](X, Z, ω) = p(X, Z, t) e−iωt dt, p(X, Z, t) = TF[p](X, Z, ω) eiωt dω. (33) 2π −∞ −∞ Now using the Fourier transform of the linear Euler’s equation ρ0 ∂VX /∂t = −∂p/∂X, the dimensional form of the impedance relation is then iωρ0 TF[p](X = 0, Z, ω) = ZS (ω, θi )
∂TF[p] (X = 0, Z, ω) + O(M) ∂X
(34)
The omitted terms in Eq. (34) emanate from quadratic nonlinearities of the Euler equation. In terms of the dimensionless variables introduced in the previous section for the description of the caustic, however, an alternative definition of Fourier transformation must be given: ∞ ∞ ˆ ˆ TF[pˆ i ](ω, ˆ u) = pˆ i (ξ, u) e−iωξ dξ, TF [pˆ r ](ω, ˆ v) = pˆ r (η, v) e−iωη dη, (35) −∞
−∞
with the dimensionless angular frequency ωˆ given by ωˆ = ω/ω0 , where ω0 is the characteristic frequency (typically the inverse of the signal duration). The inverse transforms are defined in a manner completely analogous to that given immediately above for dimensional variables. The relationship between these different definitions of Fourier transformation is obtained below, directly from the definition: TF[pi ](X, Z, ω) =
1 TF[pˆ i ](ω, ˆ u) exp[−iω(kZ ˆ cos Ω ω0 −kX sin Ω − 4ε3 k2 XZ cos 2Ω − 2ε3 sin 2Ω(Z2 − X2 ))],
TF [pr ](X, Z, ω) =
1 TF[pˆ r ](ω, ˆ v) exp[−iω(kZ ˆ cos Ω + kX sin Ω + 4ε3 k2 XZ cos 2Ω ω0 − 2ε3 sin 2Ω(Z2 − X2 ))].
(36)
From Eq. (36) and definition Eq. (20) of the space variables u and v it is easy to calculate: ∂TF[pi ] ωˆ ˆ exp[−iω(kZ sin Ω TF[pˆ i ](ω, ˆ u = 2 sin ΩZ) ˆ cos Ω (X = 0, Z, ω) = +ik ∂X ω0 −2ε3 sin 2ΩZ2 )] + O(ε), ∂TF[pr ] ωˆ ˆ exp[−iω(kZ (X = 0, Z, ω) = −ik sin Ω TF[pˆ r ](ω, ˆ v = 2 sin ΩZ) ˆ cos Ω ∂X ω0 −2ε3 sin 2ΩZ2 )] + O(ε)
(37)
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so that the ground reflected field is then found to be related to the incident one simply by a relation identical to the case of a plane wave reflection: TF[pˆ r ] = r(ω, Ω)TF[pˆ i ] =
ζS (ω, Ω) sin Ω − 1 TF[pˆ i ]. ζS (ω, Ω) sin Ω + 1
(38)
From a practical point of view, Eq. (38) is the main result of the present study: even in the nonlinear case, at the ground level, the impedance effects occur simply through a reflection coefficient identical to the linear, plane wave case. This result is not so surprising: the analysis of the previous section showed that the incident and reflected field are decoupled because they propagate in different directions, so that their nonlinear interactions are not cumulative and remain negligibly small. As a consequence, they are coupled only by the linear impedance boundary condition at the ground level. At the leading order, the focused field propagates similarly to a plane wave, diffraction effects occuring only at the higher order ε. Therefore, finite impedance effects occur at the leading order through the usual linear reflection coefficient Eq. (38). This allows a practical implementation in a sonic boom code. The incident field satisfying the nonlinear Tricomi Eq. (26) and its associated boundary conditions (Eqs. (27)–(29)) has to be solved numerically [2]. Then it is Fourier transformed, and the total field is recovered by applying the usual reflection (Eq. (38)), depending on the surface properties. Examples of such simulations will be presented in the next section. In the rigid case (ζ S = ∞) we recover the usual pressure doubling frequently used empirically for sonic boom. Finally, that section is completed by noting that, when nonlinear effects are omitted, a linear solution can be found in the frequency domain depending on the ingoing time waveform F. It is well known for the incident field [19]: √ pi (u, ξ) = TF−1 { 2π(1 + i sgn(ω))| ˆ ω| ˆ 1/6 TF(F )(ω)Ai(−| ˆ ω| ˆ 2/3 u)}. (39) Straightforwardly, a similar solution satisfying boundary conditions Eqs. (30), (31) and (38) is given for the reflected field by √ pr (v, η) = TF−1 { 2πr(ω, Ω)(1 + i sgn(ω))| ˆ ω| ˆ 1/6 TF(F )(ω)Ai(−| ˆ ω| ˆ 2/3 v)}. (40) Though for sonic boom, solution Eq. (39) is singular because incident shock waves are infinitely amplified and nonlinearities are intrinsically necessary to get finite amplitude signals, Eqs. (39) and (40) may nevertheless be useful to quantify the influence of finite impedance. The present study is limited to a linear impedance relation Eq. (32). The question arises of the validity of this assumption. A detailed study [20] based on Forchheimer’s nonlinearity of the ground impedance showed that for typical outdoor values, the ground impedance varies only a few percents for sound pressures up to several hundreds of Pa, as they can be reached by focused booms. Given the fact that surface absorption is not the dominant effect as will be shown by simulations in the following sections, we can conclude that nonlinearities of the ground acoustical properties can be considered negligible.
4. Reflection off a porous ground surface We now proceed to the numerical calculation of the incident and reflected pressure fields, at the ground level, when the incoming signal is an N-wave of length τ = 0.27 s, which is the typical duration of a Concorde sonic boom. The grazing angle Ω of the caustic above the ground is chosen equal to 20◦ , typical for an aircraft accelerating in standard atmosphere (sound speed at the ground level c0 = 340 m/s, at the supersonic flight altitude cfl = 300 m/s) such that ground track boom focusing occurs around Mach 1.2. The value 20◦ is then an immediate consequence of the Snell–Descartes law, recalling that the rays are launched from the aircraft perpendicular to the Mach cone [12]. To calculate the complex impedance of a porous ground we use Attenborough’s four-parameter model [13], the four parameters being the porosity of connected air-filled pores (Ωp ), the flow resistivity (σ), the pore-shape factor (sp ), and the tortuosity (T). Flow resistivity is a good indicator of the impedance, so it is this parameter that we will
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Fig. 2. Incident (thick continuous line) and reflected pressure waveforms for different flow resistivities: 300,000 (thin continuous line), 100,000 (dotted line), 30,000 (dashed line) and 10,000 (dash-dotted line) at the point of maximum amplification and for a nonlinear parameter µ = 0.025.
choose to gauge the differences between rigid and soft grounds. A very large value of this parameter corresponds to a surface which is almost rigid and, conversely, a smaller value indicates that the surface is more absorbent. Here, we will vary the resistivity between values of 10,000 Pa s/m2 (absorbent ground) and 1,000,000 Pa s/m2 (almost rigid ground), representative for a variety of outdoor terrains. The values of the other parameters are kept constant, and are as follows: Ωp = 0.5, sp = 0.75, T = 1.5, all being realistic values. The incident field is computed by using the numerical software developed by Marchiano et al. [2]. To examine the role of nonlinearities, two values of the nonlinear parameter µ have been selected: 0.025 and 0.1. Both of them are realistic for sonic boom focusing due to an aircraft transsonic acceleration [4]. Then, the point of maximum amplification is selected (slightly shifted away from the geometrical acoustics because of nonlinear effects), and the associated reflected boom is computed using the standard linear reflection coefficient Eq. (38). That operation is finally repeated for the different flow resistivities. The results are presented in Figs. 2–6.
Fig. 3. Same as Fig. 2 zoomed on first peak.
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Fig. 4. Same as Fig. 2 zoomed on second peak.
For the case µ = 0.025, Fig. 2 presents the incident (thick continuous line) and reflected pressure waveforms for different flow resistivity: 300,000 (thin continuous line), 100,000 (dotted line), 30,000 (dashed line) and 10,000 (dash-dotted line). The incident pressure is also equal to the reflected one for a rigid surface. The first peak of the signal is zoomed in Fig. 3, and the second one in Fig. 4. The same cases are presented for the more nonlinear case µ = 0.1 in Fig. 6. Fig. 2 shows that the overall signal appearance does not vary much with ground absorption. This was to be expected, as most energy of sonic boom is contained in the low, almost infrasonic part of the spectrum (around a few Hz) which is very insensitive to ground characteristics. Higher frequencies are contained in the shock waves and, once amplified at the caustics, in the two peaks of the waveforms, each corresponding to one shock of the N-wave. In Fig. 3, detailing the local shape of the first peak, the influence of ground absorption is much more visible. As expected, the more absorbent the ground, the higher the peak reduction. Also visible is the increase in the rise time of the signal, e.g. the time necessary for the signal to reach its peak. This parameter is known to be a key one for
Fig. 5. Same as Fig. 2 but now with µ = 0.1.
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Fig. 6. Ratio of the maximum overpressure of the reflected to the incident field as a function of flow resistivity for two values of the nonlinear parameter µ.
estimating sonic boom annoyance. It is here no longer than 5 ms. Let us note however that the increase in the rise time is not so spectacular compared to the one observed for grazing unfocused boom near the geometrical cut-off [18]. Indeed, in that last case, the rigid case leads to a perfect shock, and ground absorption, even relatively small, induces a finite rise time of the order of one or several tens of ms. Here, as propagation is not so grazing, the influence of ground absorption is not so pronounced. Moreover, the incident peak is not a shock wave. This is related to the transsonic character of the nonlinear Tricomi equation. Indeed for the point of maximum amplification, the first peak is very close to the sonic line u = µpi separating the hyperbolic and elliptic domains, but it is located slightly inside the elliptic domain. No shock can exist there, and the incident first peak is slightly rounded, even for the not absorbed incident field as can be seen in Fig. 3, with already a finite rise time. On the contrary, because of the complex shape of the sonic line, the rear part of the pressure waveform remains in the hyperbolic domain. Shock waves do exist there, as can be seen in Fig. 4 zooming on that part of the signal, where the ingoing tail shock of the N-wave shortly precedes the outgoing peak. Clearly here, the variations of the incident signal are sharper compared to Fig. 3 and rise times are much shorter. For the reflected signals, ground absorption plays a more significant role, smoothing the shock wave and blurring together the shock and the peak. Fig. 5 displays the same content as Fig. 2, but now for a more nonlinear case (µ = 0.1). The general trends are the same: the influence of surface absorption is mostly felt by the two peaks of the pressure waveforms, leading to a decrease of the peak pressure and to an increase of the rise time. The main differences are due to the fact that the shape of the incident field is somewhat different from the previous case: due to higher nonlinearities, the signal duration is increased and its amplitude is somewhat reduced (with a peak slightly above 2, compared to slightly below 3 in the previous case), in agreement with Guiraud similitude [2,3]. Also the tail shock is more clearly separated from its corresponding peak. As for the previous case, the influence of ground absorption is mostly felt where the signal variations are the sharpest. Finally, a global estimation of the combined nonlinear and ground absorption effects is illustrated in Fig. 6. This one shows the ratio of the maximum overpressures of the reflected signal to the incident one, as a function of flow resistivity. For large flow resistivities, that ratio tends towards 1, the rigid limit, while it falls off for small flow resistivities. However, to get a reduction larger than 10%, a flow resistivity smaller than 100,000 Pa s/m2 is required, which is already quite an absorbing ground. Also noticeable is the difference between the two cases, the influence of ground absorption being larger in the most linear case. This behaviour can be explained according to the detailed previous analysis, according to which absorption is more efficient on sharpest signals. For weakly nonlinear cases,
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Fig. 7. Incident (continuous line) and reflected (dotted line) pressure waveforms for different sea wave heights h in meter and for a nonlinear parameter µ = 0.04 at the point of maximum amplification. Zoom on the first peak.
the shape of the signal remains closer to the singular linear one with sharp peaks, while these one are smoother in the more nonlinear case. Therefore, ground absorption turns out to be more efficient in the weakly nonlinear case.
5. Reflection off a rough sea surface Boulanger and Attenborough have recently put forth a model designed to obtain the effective impedance spectra for rough seas [14]. These spectra can be incorporated into our propagation code in such a way that the reflected field resulting from an incident caustic arriving at the sea surface in near-grazing conditions can be calculated in much
Fig. 8. Incident (continuous line) and reflected (dotted line) pressure waveforms for different sea wave heights h in meter and for a nonlinear parameter µ = 0.04 at the point of maximum amplification. Zoom on the second peak.
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the same way as we have previously done with ground impedance. Thus, we now replace Attenborough’s ground impedance model [13], with its rough seas counterpart in our calculations, which otherwise remain unchanged apart from the angle of incidence, which is now Ω = 13◦ . We shall take the wave height h (m) as the parameter which mainly determines the different impedance spectra, in the same role the ground resistivity assumed in the ground impedance model. Below, we present several figures illustrating the differences between the incident field resulting from nonlinear propagation with µ = 0.04, and various reflected fields corresponding to different wave heights. It will be seen that these gaps between the fields are not negligible. As for ground reflection, the main effects are localized around the bow and tail peaks of the signal as illustrated in Figs. 7 and 8, respectively, for different values of the wave height ranging from 25 to 7.5 cm. For the largest sea wave heights, the difference between the fields amounts to more than one fourth of the amplitude of the front peak (0.2803), which indicates that the effect of impedance in the case of a rough sea surface is comparable to the case of the softest ground investigated. 6. Conclusion The present study has shown that both the incident and the reflected fields are solutions of two uncoupled nonlinear Tricomi equations. This decoupling occurs because the two fields propagate in different directions, at different angles to the ground, and thus, there is no cumulative interaction to be taken into account, as they would need to be in the case of plane waves. The coupling between the incident and the reflected fields is expressed only through a ground impedance condition, which is generally linear. At the surface level, the total field is then expressed in a linear form as the sum of the incident field (computed nonlinearly) and a reflected field obtained from the product of the incident field and a reflection coefficient identical to the classical linear reflection coefficient for a plane wave in similar circumstances. For waves reflected off the ground, the effects of ground impedance may be noticeable, especially in the case of a soft ground. The ground absorption acts primarily onto the peaks of the focused boom, where the high frequency part of the spectrum is localized, thus reducing their amplitude and slightly increasing the rise time. As the effect of ground absorption is most sensitive on the sharpest parts of the waveforms, amplitude reduction is enhanced in the weakly nonlinear case, where the peaks are sharper. That behaviour is similar for reflection off a sea rough surface. However, in all investigated cases, the amplitude reduction of the reflected field remains limited to no much more than 25%. Hence, from a perception point of view, no significant mitigation of superboom is likely to be expected from passive ground or sea surface absorption. The present study is limited to non-grazing caustics. For the focused boom produced by an accelerating supersonic aircraft, the caustic would be grazing at the extremities of the focusing crescent, a region matching continuously the superboom with the shadow zone. In this complex region, we expect absorption effects to be enhanced by grazing propagation, but that process remains to be explored in details. Acknowledgment This investigation has been carried out under a contract awarded by the European Commission, SOnic Boom European Research programme (“SOBER”) (contract no. G4RD-CT-2000-00398). References [1] Th. Auger, F. Coulouvrat, Numerical simulation of sonic boom focusing, AIAA J. 40 (2002) 1726–1734. [2] R. Marchiano, F. Coulouvrat, R. Grenon, Numerical simulation of shock waves focusing at fold caustics, with application to sonic boom, J. Acoust. Soc. Am. 114 (2003) 1758–1771.
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[3] J.-P. Guiraud, Acoustique g´eom´etrique, bruit balistique des avions supersoniques et focalisation, J. M´ecanique 4 (1965) 215–267 (in French). [4] Th. Auger, Mod´elisation et simulation num´erique de la focalisation d’ondes de choc acoustiques en milieu en mouvement. Application a` la focalisation du bang sonique en acc´el´eration, th`ese de doctorat de l’Universit´e Pierre et Marie Curie, Paris, 2001, p. 6 (in French). [5] R. Thom, Stabilit´e structurelle et morphogen`ese, Benjamin, Reading, 1972 (in French) (English translation: Structural stability and morphogenesis, Benjamin, Reading, 1975). [6] M.V. Berry, Singularities in waves and rays, in: R. Balian, M. Kleman, J.P. Poirier (Eds.), Physics of Defects, North-Holland, Amsterdam, 1981, pp. 453–543. [7] Yu.A. Kravtsov, Yu.I. Orlov, Caustics, Catastrophes and Wave Fields, Springer-Verlag, Berlin, 1993. [8] J.F. Nye, Natural Focussing and Fine Structure of Light, Institute of Physics Publishing, 1999. [9] R. Marchiano, J.-L. Thomas, F. Coulouvrat, Experimental simulation of supersonic superboom in a water tank: nonlinear focusing of weak shock waves at a fold caustic, Phys. Rev. Lett. 91 (18) (2003) 1–4, 184301. [10] J.-C. Wanner, J. Vall´ee, C. Vivier, C. Th´ery, Theoretical and experimental studies of the focus of sonic booms, J. Acoust. Soc. Am. 52 (1972) 1–32. [11] M. Downing, N. Zamot, C. Moss, D. Morin, E. Wolski, S. Chung, K.J. Plotkin, D. Maglieri, Controlled focused sonic booms from manoeuvreing aircraft, J. Acoust. Soc. Am. 104 (1998) 112–121. [12] F. Coulouvrat, Advances in sonic boom modelling: a review, Acta Acustica united with Acustica, submitted for publication. [13] K. Attenborough, Ground parameter information for propagation modeling, J. Acoust. Soc. Am. 92 (1992) 418–427. [14] P. Boulanger, K. Attenborough, Effective impedance spectra for predicting rough sea effects on atmospheric impulsive sounds, J. Acoust. Soc. Am. 117 (2005) 751–762; See also P. Boulanger, K. Attenborough, Effective impedance of rough sea surfaces, in: 7`eme Congr`es Franc¸ais d’Acoustique/30, Deutsche Jahrestagung f¨ur Akustik DAGA, Strasbourg, France, March 22-25, 2004, 2004, pp. 1057–1058. [15] A.D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, Acoustical Society of America, New York, 1989. [16] M.H. Holmes, Introduction to Perturbation Methods, Springer-Verlag, New York, 1995. [17] V.P. Kuznetsov, Equations of nonlinear acoustics, Sov. Phys. Acoust. 16 (1970) 467–470. [18] F. Coulouvrat, Sonic boom in the shadow zone: a geometrical theory of diffraction, J. Acoust. Soc. Am. 111 (2002) 499–508. [19] P.M. Gill, A.R. Seebass, Nonlinear acoustic behavior at a caustic: an approximate analytical solution, AIAA Paper 73-1037, 1973. [20] K. Attenborough, Nonlinear properties of ground, Technical Note TN1, Sonic Boom European Research programme (“SOBER”, European contract G4RD-CT-2000-00398), 2003.
Reflection of caustics and focused sonic booms Pablo Luis Rend´on ∗ , Franc¸ois Coulouvrat Laboratoire de Mod´elisation en M´ecanique, UMR CNRS 7607, Universit´e Pierre et Marie Curie, case 162, 4 Place Jussieu, 75252 Paris Cedex 05, France Received 23 November 2004; received in revised form 2 February 2005; accepted 8 February 2005 Available online 5 March 2005
Abstract Sonic boom, a community acceptance problem for the future development of civil high speed aircraft, is most intense (the so-called “superboom”) when focussed during the aircraft transsonic acceleration from Mach 1 to cruise speed. The ground reflection of superboom is studied. It is shown that the incident and the reflected fields are solutions of two uncoupled nonlinear Tricomi equations. The coupling between the two fields is expressed straightforwardly only through a linear ground impedance condition. For waves reflected off the ground or off rough sea surface, the effects of ground impedance may be noticeable, especially in the case of a soft ground. Amplitude reduction is also enhanced in the weakly nonlinear case, where the peaks are sharper. That behaviour is similar for reflection off a sea rough surface. However, in all investigated cases, the amplitude reduction of the reflected field remains limited. Hence, from a perception point of view, no significant mitigation of superboom is likely to be expected from passive absorption. © 2005 Elsevier B.V. All rights reserved. Keywords: Nonlinear acoustics; Sonic boom; Focusing; Caustics; Ground effects; Impedance
Introduction Sonic boom is a community acceptance problem that remains to be overcome for a supersonic overland flight, a key point for the future development of any profitable and green civil high speed aircraft. The most intense sonic boom, the so-called superboom due to sound focusing occuring during the aircraft transsonic acceleration from Mach 1 to cruise speed, cannot be avoided by realistic maneuvers. For present aircraft configurations, it leads to an amplification of ground pressures up to two to three times the unfocused boom shock strength. To comply with a future international regulation on sonic boom, an accurate prediction of the level of focused booms is required. A new algorithm has been recently designed [1,2] that is numerically solving the nonlinear Tricomi [3,4] ∗ Corresponding author. Present address: Centro de Ciencia Aplicada y Desarrollo Tecnol´ ogico, Universidad Nacional Aut´onoma de M´exico, Ciudad Universitaria, M´exico, D.F. 04510, Mexico. Tel.: +5255 56228627x1218; fax: +5255 56228675. E-mail addresses: [email protected] (P.L. Rend´on), [email protected] (F. Coulouvrat).
0165-2125/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.wavemoti.2005.02.002
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equation modelling locally the nonlinear sound field around a fold caustic (the simplest caustic in the hierarchy of catastrophe theory [5–8]). Laboratory-scale experimental simulations [9] provide quantitative validation of the model, in agreement with outdoor recordings during test flights [10,11]. However, the present models consider only caustics in the free field, while sonic boom annoyance is mostly at the ground or sea level. Until now, most calculations of the ground total field simply consider it to be twice the incident field on the ground or sea surface. While clearly convenient, the accuracy of this method would only seem guaranteed in the case of purely linear propagation and a rigid reflection surface. However, sonic boom shock waves are nonlinear, and numerical simulations show that absorbing properties of the ground may significantly influence sonic boom, mostly by increasing the rise time [12]. The question therefore remains open to determine the influence of ground properties on superboom, which amounts to modelling the problem of nonlinear reflection of caustics over a surface of finite impedance. In the present paper we propose an analytical model for the aforementioned reflection, both in the linear and the nonlinear cases, arrived at through a multiple scales method. Firstly, the geometry of the problem is presented before recalling the linear solution for the free field caustic described in the frequency domain in terms of Airy functions. In a second part, the caustic reflection is modelled in the nonlinear regime by the method of multiple scales, the mirror reflection in the linear, rigid case being used as an “Ariane thread” to sort out the intricate geometry of the problem. Despite this complexity, the result is outstandingly simple: even in the nonlinear case, both incident and reflected field satisfy two independent nonlinear Tricomi equations, the reflected field being coupled to the incident one only through the boundary condition at the ground or sea surface. This one is explicited in the third section for a surface with impedance, thus showing that the ground pressure field can be deduced from the nonlinear incident one simply by applying a linear reflection coefficient. Finally, examples of numerical simulations are performed to quantify the influence of ground porous absorption [13] or sea surface roughness [14]. 1. The geometry of the caustic Consider the two-dimensional problem of a fold caustic in a homogeneous, inviscid fluid with ambient wave speed c0 and density ρ0 . We only consider here the case of a concave caustic. This caustic arrives at a reflecting surface, either the ground or the sea surface, at an angle Ω; the sea or ground are characterised by complex impedance functions particular to each case, which will be discussed later in Sections 4 and 5. Locally, the caustic is be considered to be, essentially, a parabola, with the origin set at the point where the caustic runs into the ground or sea surface. We take an X-axis normal to the reflecting surface and pointing towards the fluid, and a Z-axis which runs along this same surface with the same orientation as sound propagation as is illustrated in Fig. 1. The origin is chosen at the contact point of the caustic with the ground.
Fig. 1. Geometry of the caustic reflection.
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For this particular geometry, the form of the incident pressure field in the vicinity of a caustic is given by Pierce [15] in the linear and high frequency approximation: pi (x, δ, t) ≈ K exp{i[ω0 t − kz − 4ε3 k2 xz + O(ε6 )]} Ai[2εkx + O(ε4 )],
(1)
with ω0 is the angular frequency, k = ω0 /c0 the wave number and Ai is the Airy function, which is the generic function describing the diffraction catastrophe for fold caustic according to catastrophe theory [5–8]. The x-axis points towards the direction normal to the caustic and the shadow zone (therefore towards ground as the caustic is concave) and the z-axis is tangent to the caustic, oriented along acoustical rays. Introducing the radius of curvature R of the caustic at the origin, the parameter ε = 1/(4kR)1/3 is a small diffraction parameter associated to the order of magnitude of diffraction effects near the caustic. In the coordinate system adapted to ground reflection x = −Z sin Ω − X cos Ω, and z = −Z cos Ω − X sin Ω, Eq. (1) now runs: ˜ Z, ˜ t) ≈ K exp{i[ω0 t − Z ˜ cos Ω + X ˜ sin Ω + 4ε3 cos 2Ω X ˜Z ˜ + 2ε3 sin 2Ω(Z ˜2 −X ˜ 2) pi (X, ˜ sin Ω + X ˜ cos Ω) + O(ε4 )], +O(ε6 )]} Ai[−2ε(Z
(2)
˜ = kX and Z ˜ = kZ have been scaled according to wavelength. where the variables X When considering reflection off a rigid ground, we may use the method of images to obtain the reflected pressure field. This is typically done by adding the contribution of a caustic placed in such a way as to be the exact mirror image with respect to the ground of the initially given incident caustic. To each ray tangent to the original caustic there will then correspond a single ray tangent to the mirror-image caustic that will satisfy the reflection condition at the ground, as is seen in Fig. 1. ˜ Z, ˜ t), The contribution to the pressure field by the original caustic, representing the incident rays, is given by pi (X, ˜ ˜ ˜ ˜ while the contribution of the image caustic is given by pr (X, Z, t) = pi (−X, Z, t). The total pressure field in the vicinity of the origin is then: ˜ Z, ˜ t) = pi (X, ˜ Z, ˜ t) + pi (−X, ˜ Z, ˜ t). p(X,
(3)
2. The method of multiple scales and the field equation The obtention of a governing equation for the pressure field in the vicinity of a caustic in free space relies on the correct choice of scalings using the nondimensional parameter ε to arrive at a compact form of this equation. Considering the reflection of this wave field off a certain surface makes things much less straightforward than in that previous case. Not only must we deal with the fields associated with incident rays and those reflected off the ground, but, more importantly, the arbitrary introduction of this boundary affects the geometry of the problem in ˜ and Z, ˜ as it was such a way that it is no longer possible to find natural scalings for each of the newly defined axes, X for the caustic in free space. In fact, attempts to find such scalings indicate that a multiple scales approach [16] is required to obtain governing equations unaffected by the influence of secular terms. In this section we will then use the method of multiple scales to obtain the governing equation for the pressure field in the vicinity of the caustic, starting from the nonlinear wave equation [17]: ∂2 Φ ∂ 1 B ∂Φ 2 2 2 , (4) − c0 ∆Φ = + (∇Φ) ∂t 2 ∂t c02 2A ∂t the gradient operator and is with Φ is the acoustical potential, B/A the parameter of nonlinearity of the fluid, ∇ the Laplacian. We assume that two scales are necessary to describe variations along the direction normal to the ground, X. The ˜ is associated solely to wavelength, while the second, X, ˆ will first of these, which we have already introduced, X, be scaled also according to the small parameter ε we have previously introduced. As we have seen in the previous
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section, the linear field associated with rays reflected off a rigid ground may be obtained from the incident field simply by replacing X by −X. When choosing the form of the phase, however, this poses a difficulty since a term proportional to X is present here, and should thus change sign depending on whether it is associated to the incident or the reflected field. We will then initially omit any terms proportional to X from the phase term, and reinstate them later when it will seem natural to do so. We start then by writing the phase as ˜ cos Ω + 2ε3 sin 2Ω(Z ˜2 −X ˜ 2 ), τˆ = ω0 t − Z
(5)
and the rest of the variables as ˆ = εkZ, Z
˜ = kX, X
ˆ = εkX, X
(6)
which is sufficient to express the total field in the linear, rigid case as ˜ Z, ˜ t) ≈ K exp[ˆτ + X ˜ sin Ω + 4ε cos 2Ω X ˆ Z] ˆ Ai[−2(Z ˆ sin Ω + X ˆ cos Ω)] p(X, ˜ sin Ω − 4ε cos 2Ω X ˆ Z] ˆ Ai[−2(Z ˆ sin Ω − X ˆ cos Ω)]. +K exp[ˆτ − X
(7)
Notice that it is not necessary to introduce two scales for the Z coordinate, as the fast variations in this direction are included in the part of the phase variable, Eq. (5) which is invariant under mirror reflection. In terms of these variables, we obtain the following relations: ∂ ∂ = ω0 , ∂t ∂ˆτ ∂ ∂ ∂ ˆ ∂ , = k −cos Ω + ε + 4ε2 k sin 2ΩZ ˆ ∂Z ∂ˆτ ∂ˆτ ∂Z ∂ ∂ ∂ ˆ ∂ , =k +ε − 4ε2 sin 2ΩX ˜ ˆ ∂X ∂ˆτ ∂X ∂X
(8) (9) (10)
ˆ in order to include the maximum number of terms of order up to where we have chosen to write ∂/∂X in terms of X ε2 in the Laplacian, which is then given by 2 2 2 ∂ Φ ∂ Φ ∂2 Φ −2 2 ∂ Φ + cos Ω 2 + 2ε − cos Ω k Φ = ˜2 ˜ X ˆ ˆ ∂ˆτ ∂X ∂X∂ ∂ˆτ ∂Z 2 ∂2 Φ ∂2 Φ ∂2 Φ 2 ∂ Φ ˆ ˆ +ε + O(ε3 ). + − 8 sin 2Ω X (11) + Z cos Ω 2 ˜ ˆ2 ˆ2 ∂ˆτ ∂ˆτ ∂X ∂X ∂Z The nonlinear terms are then rewritten as 2 2 1 B ∂Φ 2 B ∂Φ ∂Φ 2 2 2 + O(ε). + (∇Φ) = k + + cos Ω 2 ˜ 2A ∂ˆτ ∂X c0 2A ∂t
(12)
ˆ 0 where U0 is the amplitude of the acoustical velocity, and assume that We scale the potential by Φ = U0 c0 Φ/ω nonlinear effects are sufficiently small so that βM = U0 /c0 = O(ε2 ) where M is the acoustical Mach number and β = 1 + B/2A. This assumption is well satisfied for sonic boom applications: for a supersonic transport one has typically unfocused boom at the ground level of duration 0.2–0.3 s (= 1/ω0 ) and amplitude up to 120 Pa, thus leading to a parameter βM of the order of 10−3 , while the parameter ε2 would be about 4 × 10−3 for the standard atmosphere (with R about 100 km) and hence µ equal to 0.06. However, for realistic situations, the radius of curvature R is highly sensitive to meteorological conditions and can fluctuate between a few tens to several thousands of kilometers.
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We now propose a power series expansion of the form: ˆ 1 (ˆτ , X, ˆ =Φ ˆ 0 (ˆτ , X, ˆ 2 (ˆτ , X, ˜ X, ˆ Z) ˆ + εΦ ˜ X, ˆ Z) ˆ + ε2 Φ ˜ X, ˆ Z) ˆ + ···, Φ
(13)
and, in the context of a multiple-scales expansion, we seek to eliminate secular terms as they appear when matching terms of the same order in ε. We start by equating all terms of order ε0 to zero. The first equation we thus obtain is simply the wave equation: sin2 Ω
ˆ0 ˆ0 ∂2 Φ ∂2 Φ − = 0, ˜2 ∂ˆτ 2 ∂X
(14)
ˆ 0i (ξ, X, ˆ 0r = Φ ˆ 0r (η, X, ˆ 0i = Φ ˆ Z), ˆ with ξ = τˆ + X ˜ sin Ω, and Φ ˆ Z), ˆ with η = τˆ − X ˜ sin Ω; which has solutions Φ these solutions correspond, as indicated by the subindices, to the incident and reflected fields, respectively. Now, as the distinction between incident and reflected fields has become explicit, we reintroduce the terms which we omitted from the phase τ˜ , with the appropriate signs, as follows: ˜ sin Ω + 4εX ˆZ ˆ cos 2Ω, ξ = τˆ + X
˜ sin Ω − 4εX ˆZ ˆ cos 2Ω. η = τˆ − X
(15)
Notice that because the extra terms are of order ε only, Eq. (14) of order 1 and its solutions remain unchanged. The Kuznetsov equation at the order ε can be written: ˆ1 ˆ 0i ˆ 0r ˆ 0i ˆ 0r ∂2 Φ ∂ ∂Φ ∂Φ ∂Φ ∂ ∂Φ 2 = sin Ω sin Ω − cos Ω − + cos Ω , (16) 2 sin Ω ˆ ˆ ˆ ˆ ∂ξ∂η ∂ξ ∂η ∂X ∂Z ∂X ∂Z and integrating twice:
ˆ 0i ˆ 0i ∂Φ ∂Φ − cos Ω 2 sin ΩΦ1 = η sin Ω ˆ ˆ ∂X ∂Z
2
ˆ 0r ˆ 0r ∂Φ ∂Φ − ξ sin Ω + cos Ω ˆ ˆ ∂X ∂Z
ˆ 1i (ξ)+Φ ˆ 1r (η)). + 2 sin2 Ω(Φ (17)
It is at this point that the flexibility afforded by the introduction of multiple scales becomes apparent: it is possible to prevent two secular terms from appearing above, by requiring that ˆ 0i ˆ 0i ∂Φ ∂Φ − tan Ω = 0, ˆ ˆ ∂Z ∂X
and
ˆ 0r ˆ 0r ∂Φ ∂Φ + tan Ω = 0. ˆ ˆ ∂Z ∂X
(18)
Thus, we see that the solutions now have the form: ˆ 0i (ξ, u) ˆ 0i = Φ Φ
ˆ 0r = Φ ˆ 0r (η, v), and Φ
(19)
with ˆ sin Ω + X ˆ cos Ω) = −εkx, u = 2(Z
and
ˆ sin Ω − X ˆ cos Ω). v = 2(Z
(20)
Notice that u and v represent the distances to the planes tangent to the incident caustic and to the mirror caustic, respectively, and that, in the case of a concave incident caustic, we choose to have positive u and v both point towards the concave sides of the respective caustics, in the region occupied by acoustic rays. We point out, however, that u and v do not constitute an orthogonal coordinate system, as is shown in Fig. 1. The coefficient two is introduced for convenience. In similar fashion, to the first order we have ˆ 1i (ξ, u, v) + Φ ˆ 1r (η, u, v). ˆ1 =Φ Φ
(21)
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Rewriting the Kuznetsov Eq. (4) in terms of new variables (ξ, η, u, v) and injecting the asymptotic expansions Eqs. (13), (19) and (21) yields: 2ˆ ˆ 1i ˆ 1r ˆ 0i ˆ 0r ˆ 0i ˆ 0r ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ ∂2 Φ 2 ∂ Φ2 4 sin Ω + 4 sin Ω cos Ω + − +u +v +4 − ∂ξ∂η ∂ξ∂v ∂η∂u ∂u2 ∂v2 ∂ξ 2 ∂η2 2 2 2 ˆ 0i ˆ 0r ˆ 0i ∂Φ ˆ 0r ˆ 0r ˆ 0i ∂2 Φ M ∂ ∂Φ ∂ ∂Φ B ∂ Φ ∂Φ . = 2 β (22) +β +2 + cos 2Ω + ε ∂ξ ∂ξ ∂η ∂η 2A ∂ξ 2 ∂η ∂ξ ∂η2 Integrating twice with respect to variables (ξ, η) provides the following expression: 2 ξ 2ˆ ˆ ˆ ˆ ∂ Φ0i ∂Φ0i βM ∂Φ0i ˆ 2 = 4η −sin Ω cos Ω ∂Φ1i + dξ − u + 2 4 sin2 ΩΦ 2 ∂v ∂u ∂ξ 4ε ∂ξ −∞ ˆ 1r ∂Φ +4ξ −sin Ω cos Ω + ∂u
η
−∞
ˆ 0r ˆ 0r ∂2 Φ ∂Φ βM dη − v + 2 2 ∂v ∂η 4ε
ˆ 0r ∂Φ ∂η
2 + NST.
(23)
where the missing terms NST are non-secular terms. Now requiring that secular terms proportional to ξ and η both vanish, and integrating now in terms of variables u and v yields: 2 ξ 2ˆ ˆ 0i ˆ 0i Φ ∂ ∂ Φ ∂ Φ βM 0i + NST, ˆ 1i = v sin Ω cos ΩΦ dξ − u + 2 2 ∂ξ 4ε ∂ξ −∞ ∂u ˆ 1r = u sin Ω cos ΩΦ
η
−∞
ˆ ∂2 Φ
0r
∂v2
ˆ 0r ∂Φ βM dη − v + 2 ∂η 4ε
ˆ 0r ∂Φ ∂η
2 + NST.
The cancellation of secular terms, in this case proportional to u or v requires: 2 2 ˆ 0i ˆ 0i ˆ 0i ˆ 0i ˆ 0r ˆ 0r ∂2 Φ ∂2 Φ βM ∂ ∂Φ ∂2 Φ ∂2 Φ βM ∂ ∂Φ −u + 2 = 0, −v + 2 = 0. ∂u2 ∂ξ 2 4ε ∂ξ ∂ξ ∂v2 ∂η2 4ε ∂η ∂η
(24)
(25)
Thus, we are able to obtain uncoupled equations for the incident and reflected fields, as we expected, since interaction with the reflected waves is not, as it were, felt by the incident waves, and vice versa, even in the nonlinear case. Physically, this comes from the fact that the incident and reflected fields propagate in two different directions, so that their nonlinear interaction is not resonant, as is well known: the last term in Eq. (22) which is the only one ˆ ˆ = ∂Φi∂ξ(ξ,u) , where the incident and reflected fields do interact, is not secular. Now substituting pˆ i (ξ, u) = ∂Φi∂ˆ(ξ,u) τ and pˆ r (η, v) =
ˆ r (η,v) ∂Φ ∂ˆτ
=
ˆ r (η,v) ∂Φ ∂η ,
we may write:
∂2 pˆ i ∂2 pˆ i ∂2 (pˆ i )2 − u + µ = 0, ∂u2 ∂ξ 2 ∂ξ 2
∂2 pˆ r ∂2 pˆ r ∂2 (pˆ r )2 − v + µ = 0, ∂v2 ∂η2 ∂η2
(26)
with µ = βM/4ε2 is a nonlinear parameter measuring the relative magnitude order of nonlinear effects (βM) compared to diffraction ones (ε2 ). We thus obtain uncoupled nonlinear Tricomi equations for the incident and reflected fields in the region near the point where the caustic meets the ground. Guiraud [3] has previously shown that the Tricomi equation describes the pressure field in the vicinity of a caustic in free space, so this result is, in fact, an expected one, but is still important since it indicates that it is possible to deal with the case of the incident and the reflected fields
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separately, as the governing equations are uncoupled. The relation between the solutions to these two equations will be expressed through the boundary conditions imposed on them, and in particular an impedance condition given on the ground, as detailed in the next section. Finally, it should be noted that the present study is not applicable to the case of a grazing caustic. Indeed, for very small angles Ω, the first term in the left-hand side of Eq. (14) may turn out to be of second order ε, thus upsetting the present hierarchy of equations. Consequently, the present study is limited to grazing angles Ω larger than ε1/2 . In practice, for the focused sonic (super) boom resulting from transonic acceleration, the caustic intersection with the ground has typically the shape of a crescent. The center of this crescent is associated to focused boom emitted right below the aircraft (ground track focusing) at the smallest Mach numbers, while its extremities are grazing above the ground, and continuously match the caustics with the shadow zone [18]. Such a complex region is beyond the scope of the present study and would need further asymptotic modelling. It is to be noted that the limiting value Ω ≈ ε1/2 delimiting the validity of the present theory is not so small: for the typical value ε2 = 4 × 10−3 in the standard atmosphere, this provides a grazing angle necessarily larger than 14◦ .
3. The boundary conditions Two sets of boundary conditions are required, one set for the incoming field and another for the reflected field, so that, in total, four conditions must be given, due to the fact that the Tricomi equations obtained above are of second order for each of the two variables involved. Let us consider first the boundary conditions for the incoming field, which have the same form as the analogous expression for a free-field fold caustic [1–4]. For a transient signal, we require that the field vanishes for very large times, thus obtaining the two boundary conditions for ξ: pˆ i (ξ → ±∞, u) = 0.
(27)
Recall that the Tricomi equation is a mixed hyperbolic/elliptic equation. For u − µpi > 0, it is of hyperbolic type, and for u − µpi < 0, it is of elliptic type; thus, two different behaviours are to be expected on either side of the caustic. As in the case of a caustic in free space [1–4], on the elliptic side the field vanishes exponentially as we go away from the caustic and into the shadow zone. Since u indicates the direction normal to the caustic at the origin, this condition is here expressed as follows: pˆ i (ξ, u → −∞) −→ 0. exp
(28)
We also require that the pressure field matches the geometrical acoustics approximation far away from the caustic on the hyperbolic side of the caustic: pˆ i (ξ, u → +∞) ≈ |u|−1/4 [F (ξ − 23 u3/2 ) + G(ξ + 23 u3/2 )],
(29)
where the minus sign corresponds to F, the ingoing field before it tangents the caustic, and the plus sign corresponds to G, the outgoing field after it has tangented the caustic. In Eq. (29), only the ingoing field is known contrarily to the outgoing one. This last one can be eliminated so as to rewrite Eq. (29) under the form of a “radiation” condition involving only the incoming field [1,2,4]. Now we consider the boundary conditions for the reflected field. The condition regarding the phase is repeated for the reflected field: pˆ r (η → ±∞, v) = 0,
(30)
and so is the condition of exponential decay, applied, as before, to the field for large v, since the influence of the ground is not yet felt: pˆ r (η, v → −∞) −→ 0. exp
(31)
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However, for v positive, the influence of the ground on the propagation of the reflected rays must be taken into account to formulate the remaining condition. Assuming that the ground surface S is not rigid or impenetrable, we use a specific acoustic impedance [15] denoted ZS to characterize the interaction of the incoming acoustic rays with the ground, and which is a function only of the angular frequency ω and the angle Ω of incidence of the incoming field, and is given in this case by TF[p](X, Z, ω) , (32) ZS (ω, θi ) = c0 ρ0 ζS (ω, θi ) = − TF[VX ](X, Z, ω) X=0 where VX (X, Z, t) is the fluid velocity in the ground direction. The ratio of the specific impedance ZS to the characteristic air impedance is denoted ζ S , a dimensionless impedance which we will use in what follows. The direct and inverse Fourier transforms are defined, for dimensional, physical variables, as follows: ∞ ∞ 1 TF[p](X, Z, ω) = p(X, Z, t) e−iωt dt, p(X, Z, t) = TF[p](X, Z, ω) eiωt dω. (33) 2π −∞ −∞ Now using the Fourier transform of the linear Euler’s equation ρ0 ∂VX /∂t = −∂p/∂X, the dimensional form of the impedance relation is then iωρ0 TF[p](X = 0, Z, ω) = ZS (ω, θi )
∂TF[p] (X = 0, Z, ω) + O(M) ∂X
(34)
The omitted terms in Eq. (34) emanate from quadratic nonlinearities of the Euler equation. In terms of the dimensionless variables introduced in the previous section for the description of the caustic, however, an alternative definition of Fourier transformation must be given: ∞ ∞ ˆ ˆ TF[pˆ i ](ω, ˆ u) = pˆ i (ξ, u) e−iωξ dξ, TF [pˆ r ](ω, ˆ v) = pˆ r (η, v) e−iωη dη, (35) −∞
−∞
with the dimensionless angular frequency ωˆ given by ωˆ = ω/ω0 , where ω0 is the characteristic frequency (typically the inverse of the signal duration). The inverse transforms are defined in a manner completely analogous to that given immediately above for dimensional variables. The relationship between these different definitions of Fourier transformation is obtained below, directly from the definition: TF[pi ](X, Z, ω) =
1 TF[pˆ i ](ω, ˆ u) exp[−iω(kZ ˆ cos Ω ω0 −kX sin Ω − 4ε3 k2 XZ cos 2Ω − 2ε3 sin 2Ω(Z2 − X2 ))],
TF [pr ](X, Z, ω) =
1 TF[pˆ r ](ω, ˆ v) exp[−iω(kZ ˆ cos Ω + kX sin Ω + 4ε3 k2 XZ cos 2Ω ω0 − 2ε3 sin 2Ω(Z2 − X2 ))].
(36)
From Eq. (36) and definition Eq. (20) of the space variables u and v it is easy to calculate: ∂TF[pi ] ωˆ ˆ exp[−iω(kZ sin Ω TF[pˆ i ](ω, ˆ u = 2 sin ΩZ) ˆ cos Ω (X = 0, Z, ω) = +ik ∂X ω0 −2ε3 sin 2ΩZ2 )] + O(ε), ∂TF[pr ] ωˆ ˆ exp[−iω(kZ (X = 0, Z, ω) = −ik sin Ω TF[pˆ r ](ω, ˆ v = 2 sin ΩZ) ˆ cos Ω ∂X ω0 −2ε3 sin 2ΩZ2 )] + O(ε)
(37)
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so that the ground reflected field is then found to be related to the incident one simply by a relation identical to the case of a plane wave reflection: TF[pˆ r ] = r(ω, Ω)TF[pˆ i ] =
ζS (ω, Ω) sin Ω − 1 TF[pˆ i ]. ζS (ω, Ω) sin Ω + 1
(38)
From a practical point of view, Eq. (38) is the main result of the present study: even in the nonlinear case, at the ground level, the impedance effects occur simply through a reflection coefficient identical to the linear, plane wave case. This result is not so surprising: the analysis of the previous section showed that the incident and reflected field are decoupled because they propagate in different directions, so that their nonlinear interactions are not cumulative and remain negligibly small. As a consequence, they are coupled only by the linear impedance boundary condition at the ground level. At the leading order, the focused field propagates similarly to a plane wave, diffraction effects occuring only at the higher order ε. Therefore, finite impedance effects occur at the leading order through the usual linear reflection coefficient Eq. (38). This allows a practical implementation in a sonic boom code. The incident field satisfying the nonlinear Tricomi Eq. (26) and its associated boundary conditions (Eqs. (27)–(29)) has to be solved numerically [2]. Then it is Fourier transformed, and the total field is recovered by applying the usual reflection (Eq. (38)), depending on the surface properties. Examples of such simulations will be presented in the next section. In the rigid case (ζ S = ∞) we recover the usual pressure doubling frequently used empirically for sonic boom. Finally, that section is completed by noting that, when nonlinear effects are omitted, a linear solution can be found in the frequency domain depending on the ingoing time waveform F. It is well known for the incident field [19]: √ pi (u, ξ) = TF−1 { 2π(1 + i sgn(ω))| ˆ ω| ˆ 1/6 TF(F )(ω)Ai(−| ˆ ω| ˆ 2/3 u)}. (39) Straightforwardly, a similar solution satisfying boundary conditions Eqs. (30), (31) and (38) is given for the reflected field by √ pr (v, η) = TF−1 { 2πr(ω, Ω)(1 + i sgn(ω))| ˆ ω| ˆ 1/6 TF(F )(ω)Ai(−| ˆ ω| ˆ 2/3 v)}. (40) Though for sonic boom, solution Eq. (39) is singular because incident shock waves are infinitely amplified and nonlinearities are intrinsically necessary to get finite amplitude signals, Eqs. (39) and (40) may nevertheless be useful to quantify the influence of finite impedance. The present study is limited to a linear impedance relation Eq. (32). The question arises of the validity of this assumption. A detailed study [20] based on Forchheimer’s nonlinearity of the ground impedance showed that for typical outdoor values, the ground impedance varies only a few percents for sound pressures up to several hundreds of Pa, as they can be reached by focused booms. Given the fact that surface absorption is not the dominant effect as will be shown by simulations in the following sections, we can conclude that nonlinearities of the ground acoustical properties can be considered negligible.
4. Reflection off a porous ground surface We now proceed to the numerical calculation of the incident and reflected pressure fields, at the ground level, when the incoming signal is an N-wave of length τ = 0.27 s, which is the typical duration of a Concorde sonic boom. The grazing angle Ω of the caustic above the ground is chosen equal to 20◦ , typical for an aircraft accelerating in standard atmosphere (sound speed at the ground level c0 = 340 m/s, at the supersonic flight altitude cfl = 300 m/s) such that ground track boom focusing occurs around Mach 1.2. The value 20◦ is then an immediate consequence of the Snell–Descartes law, recalling that the rays are launched from the aircraft perpendicular to the Mach cone [12]. To calculate the complex impedance of a porous ground we use Attenborough’s four-parameter model [13], the four parameters being the porosity of connected air-filled pores (Ωp ), the flow resistivity (σ), the pore-shape factor (sp ), and the tortuosity (T). Flow resistivity is a good indicator of the impedance, so it is this parameter that we will
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Fig. 2. Incident (thick continuous line) and reflected pressure waveforms for different flow resistivities: 300,000 (thin continuous line), 100,000 (dotted line), 30,000 (dashed line) and 10,000 (dash-dotted line) at the point of maximum amplification and for a nonlinear parameter µ = 0.025.
choose to gauge the differences between rigid and soft grounds. A very large value of this parameter corresponds to a surface which is almost rigid and, conversely, a smaller value indicates that the surface is more absorbent. Here, we will vary the resistivity between values of 10,000 Pa s/m2 (absorbent ground) and 1,000,000 Pa s/m2 (almost rigid ground), representative for a variety of outdoor terrains. The values of the other parameters are kept constant, and are as follows: Ωp = 0.5, sp = 0.75, T = 1.5, all being realistic values. The incident field is computed by using the numerical software developed by Marchiano et al. [2]. To examine the role of nonlinearities, two values of the nonlinear parameter µ have been selected: 0.025 and 0.1. Both of them are realistic for sonic boom focusing due to an aircraft transsonic acceleration [4]. Then, the point of maximum amplification is selected (slightly shifted away from the geometrical acoustics because of nonlinear effects), and the associated reflected boom is computed using the standard linear reflection coefficient Eq. (38). That operation is finally repeated for the different flow resistivities. The results are presented in Figs. 2–6.
Fig. 3. Same as Fig. 2 zoomed on first peak.
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Fig. 4. Same as Fig. 2 zoomed on second peak.
For the case µ = 0.025, Fig. 2 presents the incident (thick continuous line) and reflected pressure waveforms for different flow resistivity: 300,000 (thin continuous line), 100,000 (dotted line), 30,000 (dashed line) and 10,000 (dash-dotted line). The incident pressure is also equal to the reflected one for a rigid surface. The first peak of the signal is zoomed in Fig. 3, and the second one in Fig. 4. The same cases are presented for the more nonlinear case µ = 0.1 in Fig. 6. Fig. 2 shows that the overall signal appearance does not vary much with ground absorption. This was to be expected, as most energy of sonic boom is contained in the low, almost infrasonic part of the spectrum (around a few Hz) which is very insensitive to ground characteristics. Higher frequencies are contained in the shock waves and, once amplified at the caustics, in the two peaks of the waveforms, each corresponding to one shock of the N-wave. In Fig. 3, detailing the local shape of the first peak, the influence of ground absorption is much more visible. As expected, the more absorbent the ground, the higher the peak reduction. Also visible is the increase in the rise time of the signal, e.g. the time necessary for the signal to reach its peak. This parameter is known to be a key one for
Fig. 5. Same as Fig. 2 but now with µ = 0.1.
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Fig. 6. Ratio of the maximum overpressure of the reflected to the incident field as a function of flow resistivity for two values of the nonlinear parameter µ.
estimating sonic boom annoyance. It is here no longer than 5 ms. Let us note however that the increase in the rise time is not so spectacular compared to the one observed for grazing unfocused boom near the geometrical cut-off [18]. Indeed, in that last case, the rigid case leads to a perfect shock, and ground absorption, even relatively small, induces a finite rise time of the order of one or several tens of ms. Here, as propagation is not so grazing, the influence of ground absorption is not so pronounced. Moreover, the incident peak is not a shock wave. This is related to the transsonic character of the nonlinear Tricomi equation. Indeed for the point of maximum amplification, the first peak is very close to the sonic line u = µpi separating the hyperbolic and elliptic domains, but it is located slightly inside the elliptic domain. No shock can exist there, and the incident first peak is slightly rounded, even for the not absorbed incident field as can be seen in Fig. 3, with already a finite rise time. On the contrary, because of the complex shape of the sonic line, the rear part of the pressure waveform remains in the hyperbolic domain. Shock waves do exist there, as can be seen in Fig. 4 zooming on that part of the signal, where the ingoing tail shock of the N-wave shortly precedes the outgoing peak. Clearly here, the variations of the incident signal are sharper compared to Fig. 3 and rise times are much shorter. For the reflected signals, ground absorption plays a more significant role, smoothing the shock wave and blurring together the shock and the peak. Fig. 5 displays the same content as Fig. 2, but now for a more nonlinear case (µ = 0.1). The general trends are the same: the influence of surface absorption is mostly felt by the two peaks of the pressure waveforms, leading to a decrease of the peak pressure and to an increase of the rise time. The main differences are due to the fact that the shape of the incident field is somewhat different from the previous case: due to higher nonlinearities, the signal duration is increased and its amplitude is somewhat reduced (with a peak slightly above 2, compared to slightly below 3 in the previous case), in agreement with Guiraud similitude [2,3]. Also the tail shock is more clearly separated from its corresponding peak. As for the previous case, the influence of ground absorption is mostly felt where the signal variations are the sharpest. Finally, a global estimation of the combined nonlinear and ground absorption effects is illustrated in Fig. 6. This one shows the ratio of the maximum overpressures of the reflected signal to the incident one, as a function of flow resistivity. For large flow resistivities, that ratio tends towards 1, the rigid limit, while it falls off for small flow resistivities. However, to get a reduction larger than 10%, a flow resistivity smaller than 100,000 Pa s/m2 is required, which is already quite an absorbing ground. Also noticeable is the difference between the two cases, the influence of ground absorption being larger in the most linear case. This behaviour can be explained according to the detailed previous analysis, according to which absorption is more efficient on sharpest signals. For weakly nonlinear cases,
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Fig. 7. Incident (continuous line) and reflected (dotted line) pressure waveforms for different sea wave heights h in meter and for a nonlinear parameter µ = 0.04 at the point of maximum amplification. Zoom on the first peak.
the shape of the signal remains closer to the singular linear one with sharp peaks, while these one are smoother in the more nonlinear case. Therefore, ground absorption turns out to be more efficient in the weakly nonlinear case.
5. Reflection off a rough sea surface Boulanger and Attenborough have recently put forth a model designed to obtain the effective impedance spectra for rough seas [14]. These spectra can be incorporated into our propagation code in such a way that the reflected field resulting from an incident caustic arriving at the sea surface in near-grazing conditions can be calculated in much
Fig. 8. Incident (continuous line) and reflected (dotted line) pressure waveforms for different sea wave heights h in meter and for a nonlinear parameter µ = 0.04 at the point of maximum amplification. Zoom on the second peak.
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the same way as we have previously done with ground impedance. Thus, we now replace Attenborough’s ground impedance model [13], with its rough seas counterpart in our calculations, which otherwise remain unchanged apart from the angle of incidence, which is now Ω = 13◦ . We shall take the wave height h (m) as the parameter which mainly determines the different impedance spectra, in the same role the ground resistivity assumed in the ground impedance model. Below, we present several figures illustrating the differences between the incident field resulting from nonlinear propagation with µ = 0.04, and various reflected fields corresponding to different wave heights. It will be seen that these gaps between the fields are not negligible. As for ground reflection, the main effects are localized around the bow and tail peaks of the signal as illustrated in Figs. 7 and 8, respectively, for different values of the wave height ranging from 25 to 7.5 cm. For the largest sea wave heights, the difference between the fields amounts to more than one fourth of the amplitude of the front peak (0.2803), which indicates that the effect of impedance in the case of a rough sea surface is comparable to the case of the softest ground investigated. 6. Conclusion The present study has shown that both the incident and the reflected fields are solutions of two uncoupled nonlinear Tricomi equations. This decoupling occurs because the two fields propagate in different directions, at different angles to the ground, and thus, there is no cumulative interaction to be taken into account, as they would need to be in the case of plane waves. The coupling between the incident and the reflected fields is expressed only through a ground impedance condition, which is generally linear. At the surface level, the total field is then expressed in a linear form as the sum of the incident field (computed nonlinearly) and a reflected field obtained from the product of the incident field and a reflection coefficient identical to the classical linear reflection coefficient for a plane wave in similar circumstances. For waves reflected off the ground, the effects of ground impedance may be noticeable, especially in the case of a soft ground. The ground absorption acts primarily onto the peaks of the focused boom, where the high frequency part of the spectrum is localized, thus reducing their amplitude and slightly increasing the rise time. As the effect of ground absorption is most sensitive on the sharpest parts of the waveforms, amplitude reduction is enhanced in the weakly nonlinear case, where the peaks are sharper. That behaviour is similar for reflection off a sea rough surface. However, in all investigated cases, the amplitude reduction of the reflected field remains limited to no much more than 25%. Hence, from a perception point of view, no significant mitigation of superboom is likely to be expected from passive ground or sea surface absorption. The present study is limited to non-grazing caustics. For the focused boom produced by an accelerating supersonic aircraft, the caustic would be grazing at the extremities of the focusing crescent, a region matching continuously the superboom with the shadow zone. In this complex region, we expect absorption effects to be enhanced by grazing propagation, but that process remains to be explored in details. Acknowledgment This investigation has been carried out under a contract awarded by the European Commission, SOnic Boom European Research programme (“SOBER”) (contract no. G4RD-CT-2000-00398). References [1] Th. Auger, F. Coulouvrat, Numerical simulation of sonic boom focusing, AIAA J. 40 (2002) 1726–1734. [2] R. Marchiano, F. Coulouvrat, R. Grenon, Numerical simulation of shock waves focusing at fold caustics, with application to sonic boom, J. Acoust. Soc. Am. 114 (2003) 1758–1771.
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